What does the phrase 'food for thought' mean?

and 'chew the cud'?

I always used "food for thought" to refer to something to think about or ponder, maybe an idea or such.

I think of "chewing the cud" as pondering or thinking about that afore-mentioned idea.

("Cud for thought"?
Or "Chewing the food-for-thought"? )

There is an interesting connection between these two expressions by way of the word ruminate. Cows and people can both be said to ruminate. When a cow does it, it is literally "chewing the cud." But when a person ruminates, he or she is metaphorically chewing over food for thought, as TIE has already put it.